ployphase decomposition
TRANSCRIPT
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1Copyright 2001, S. K. Mitra
Polyphase Decomposition
The Decomposition
Consider an arbitrary sequence {x[n]} with
az-transformX(z) given by
We can rewriteX(z) as
where
n
nznxzX][)(
1
0M
k
M
k
kzXzzX )()(
n
n
n
n
kk zkMnxznxzX ][][)(
10 Mk
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2Copyright 2001, S. K. Mitra
Polyphase Decomposition
The subsequences are called the
polyphase components of the parent
sequence {x[n]} The functions , given by the
z-transforms of , are called the
polyphase componentsofX(z)
]}[{ nxk
]}[{ nxk
)(zXk
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3Copyright 2001, S. K. Mitra
Polyphase Decomposition
The relation between the subsequences
and the original sequence {x[n]} are given
by
In matrix form we can write
]}[{ nxk
10 MkkMnxnxk ],[][
)(
)(
)(
....)( )(
M
M
M
M
M
zX
zX
zX
zzzX
1
10
111 ....
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4 Copyright 2001, S. K. Mitra
Polyphase Decomposition
A multirate structural interpretation of the
polyphase decomposition is given below
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5 Copyright 2001, S. K. Mitra
Polyphase Decomposition
Thepolyphase decomposition of an FIR
transfer function can be carried out by
inspection
For example, consider a length-9 FIR
transfer function:
8
0n
nznhzH ][)(
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6 Copyright 2001, S. K. Mitra
Polyphase Decomposition
Its 4-branch polyphase decomposition is
given by
where
)()()()()( 4
3
34
2
24
1
14
0zEzzEzzEzzEzH
210 840
zhzhhzE ][][][)(
11 51
zhhzE ][][)(1
2 62 zhhzE ][][)(
13 73
zhhzE ][][)(
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7 Copyright 2001, S. K. Mitra
Polyphase Decomposition
Thepolyphase decomposition of an IIR
transfer functionH(z) =P(z)/D(z) is not that
straight forward
One way to arrive at anM-branch polyphase
decomposition ofH(z) is to express it in the
form by multiplyingP(z) and
D(z) with an appropriately chosenpolynomial and then apply anM-branch
polyphase decomposition to
)('/)(' MzDzP
)(' zP
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8 Copyright 2001, S. K. Mitra
Polyphase Decomposition Example -Consider
To obtain a 2-band polyphase decomposition werewriteH(z) as
Therefore,
where
1
1
31
21
z
zzH )(
2
1
2
2
2
21
11
11
91
5
91
61
91
651
)31)(31(
)31)(21()(
z
z
z
z
z
zz
zz
zzzH
)()()( 2112
0 zEzzEzH
11
1
91
51
91
610
zz
zzEzE )(,)(
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9 Copyright 2001, S. K. Mitra
Polyphase Decomposition
Note:The above approach increases the
overall order and complexity ofH(z)
However, when used in certain multirate
structures, the approach may result in a
more computationally efficient structure
An alternative more attractive approach isdiscussed in the following example
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10 Copyright 2001, S. K. Mitra
Polyphase Decomposition
Example -Consider the transfer function of
a 5-th order Butterworth lowpass filter with
a 3-dB cutoff frequency at 0.5p:
It is easy to show thatH(z) can be expressedas
1 5
2 4
0.0527864 (1 )
1 0.633436854 0.0557281( )
z
z zH z
2
2
2
2
5278601
5278601
10557301
1055730
2
1
z
z
z
zzzH
.
.
.
.)(
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11 Copyright 2001, S. K. Mitra
Polyphase Decomposition
ThereforeH(z) can be expressed as
where
1
1
5278601
527860
2
11
z
zzE
.
.)(
1
1
10557301
1055730
2
10
z
zzE
.
.)(
)()()( 2112
0 zEzzEzH
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12 Copyright 2001, S. K. Mitra
Polyphase Decomposition
Note: In the above polyphase decomposition,
branch transfer functions arestable
allpass functions
Moreover, the decomposition has not
increased the order of the overall transfer
functionH(z)
)(zEi
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13 Copyright 2001, S. K. Mitra
FIR Filter Structures Based on
Polyphase Decomposition We shall demonstrate later that a parallel
realization of an FIR transfer functionH(z)
based on the polyphase decomposition canoften result in computationally efficient
multirate structures
Consider theM-branch Type I polyphasedecomposition ofH(z):
)()(1
0M
k
M
k
k zEzzH
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14 Copyright 2001, S. K. Mitra
FIR Filter Structures Based on
Polyphase Decomposition A direct realization ofH(z)based on the
Type I polyphase decomposition is shown
below
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15 Copyright 2001, S. K. Mitra
FIR Filter Structures Based on
Polyphase Decomposition The transpose of the Type I polyphase FIR
filter structure is indicated below
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16 Copyright 2001, S. K. Mitra
FIR Filter Structures Based on
Polyphase Decomposition An alternative representation of the
transpose structure shown on the previous
slide is obtained using the notation
Substituting the above notation in the Type
I polyphase decomposition we arrive at theType II polyphase decomposition:
)()(1
0)1( MM M zRzzH
10),()( 1 MzEzRM
MM
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17 Copyright 2001, S. K. Mitra
FIR Filter Structures Based on
Polyphase Decomposition A direct realization ofH(z)based on the
Type II polyphase decomposition is shown
below
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18 Copyright 2001, S. K. Mitra
Computationally Efficient
Decimators Consider first the single-stage factor-of-M
decimator structure shown below
We realize the lowpass filterH(z) using the
Type I polyphase structure as shown on the
next slide
M][nx )(zH ][nyv[n]
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19 Copyright 2001, S. K. Mitra
Computationally EfficientDecimators
Using the cascade equivalence #1 we arriveat the computationally efficient decimatorstructure shown below on the right
Decimator structure based on Type I polyphase decomposition
y[n] y[n]x[n]x[n]
][nv
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20 Copyright 2001, S. K. Mitra
Computationally Efficient
Decimators To illustrate the computational efficiency of
the modified decimator structure, assume
H(z) to be a length-Nstructure and the inputsampling period to beT= 1
Now the decimator outputy[n] in the
original structure is obtained by down-sampling the filter outputv[n]by a factor of
M
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21 Copyright 2001, S. K. Mitra
Computationally Efficient
Decimators It is thus necessary to computev[n] at
Computational requirements are thereforeN
multiplications and additions per
output sample being computed
However, asnincreases, stored signals in
the delay registers change
...,2,,0,,2,... MMMMn
)1( N
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22 Copyright 2001, S. K. Mitra
Computationally Efficient
Decimators Hence, all computations need to be
completed in one sampling period, and for
the following sampling periods thearithmetic units remain idle
The modified decimator structure also
requiresNmultiplications andadditions per output sample being computed
)1( N
)1( M
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23 Copyright 2001, S. K. Mitra
Computationally Efficient
Decimators and Interpolators However, here the arithmetic units are
operative at all instants of the output
sampling period which isMtimes that ofthe input sampling period
Similar savings are also obtained in the case
of the interpolator structure developed usingthe polyphase decomposition
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24 Copyright 2001, S. K. Mitra
Computationally EfficientInterpolators
Figures below show the computationally
efficient interpolator structures
Interpolator based onType I polyphase decomposition
Interpolator based onType II polyphase decomposition
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25 Copyright 2001, S. K. Mitra
Computationally Efficient
Decimators and Interpolators More efficient interpolator and decimator
structures can be realized by exploiting the
symmetry of filter coefficients in the case oflinear-phase filtersH(z)
Consider for example the realization of a
factor-of-3(M= 3) decimator using alength-12 Type 1linear-phase FIR lowpass
filter
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26 Copyright 2001, S. K. Mitra
Computationally Efficient
Decimators and Interpolators The corresponding transfer function is
A conventional polyphase decomposition of
H(z) yields the following subfilters:
54321 ]5[]4[]3[]2[]1[]0[)( zhzhzhzhzhhzH
11109876 ]0[]1[]2[]3[]4[]5[ zhzhzhzhzhzh
3210 ]2[]5[]3[]0[)( zhzhzhhzE
3211 ]1[]4[]4[]1[)(
zhzhzhhzE321
2 ]0[]3[]5[]2[)( zhzhzhhzE
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27 Copyright 2001, S. K. Mitra
Computationally Efficient
Decimators and Interpolators Note that still has a symmetric
impulse response, whereas is the
mirror image of These relations can be made use of in
developing a computationally efficient
realization using only 6 multipliers and 11two-input adders as shown on the next slide
)(1 zE
)(0 zE
)(2 zE
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28 Copyright 2001, S. K. Mitra
Computationally Efficient
Decimators and Interpolators Factor-of-3decimator with a linear-phase
decimation filter
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29 Copyright 2001 S K Mitra
A Useful Identity The cascade multirate structure shown
below appears in a number of applications
Equivalent time-invariant digital filter
obtained by expressingH(z) in itsL-term
Type I polyphase form
is shown below
L LH(z)x[n] y[n]
)(10L
kL
kk zEz
x[n] y[n])(0 zE